Bridging functional and anatomical neural connectivity through cluster synchronization

The dynamics of the brain results from the complex interplay of several neural populations and is affected by both the individual dynamics of these areas and their connection structure. Hence, a fundamental challenge is to derive models of the brain that reproduce both structural and functional features measured experimentally. Our work combines neuroimaging data, such as dMRI, which provides information on the structure of the anatomical connectomes, and fMRI, which detects patterns of approximate synchronous activity between brain areas. We employ cluster synchronization as a tool to integrate the imaging data of a subject into a coherent model, which reconciles structural and dynamic information. By using data-driven and model-based approaches, we refine the structural connectivity matrix in agreement with experimentally observed clusters of brain areas that display coherent activity. The proposed approach leverages the assumption of homogeneous brain areas; we show the robustness of this approach when heterogeneity between the brain areas is introduced in the form of noise, parameter mismatches, and connection delays. As a proof of concept, we apply this approach to MRI data of a healthy adult at resting state.


Note 2
This note contains the results of the application of the proposed method to functional and structural data of two additional subjects of the HNU1 dataset 1 in the Neurodata MRI Cloud database 2 .The left side of Fig. S2 shows the Ψ 1 (ℓ) index for all subjects of the dataset, color-coded according to the bottom colorbar.The index peaks, which identify the sets of selected levels L * , are marked with red dots.In the main paper, we present results for subject 0025452, which showed the highest similarity index Ψ 1 averaged across levels.Here, we present equivalent results for subjects 0025428 and 0025447, which show the second and third highest average similarity index Ψ 1 , respectively.This criterion of selection favors subjects that exhibit good agreement among clusterizations obtained from fMRI sessions.In other words, we considered subjects that exhibited a similar resting state activity, in terms of groups of synchronized brain areas, across different recordings.The right side of Fig. S2 displays the isolated Ψ 1 (ℓ) curves of the two selected subjects.According to the Ψ 2 (s) curves calculated for these subjects, we selected the X * correlation matrices as follows: the one associated with session 1 for subject 0025428, and the one associated with session 5 for subject 0025447.Fig. S3 shows the dendrograms obtained from the hierarchical clustering applied on matrices X * for both subjects, where the central levels of the sets L * (level ℓ * 18 for subject 0025428 and level ℓ * 21 for subject 0025447) are marked by red horizontal lines.Nodes on the dendrogram are labeled according to the HOA.
We remark that matrix Ξ k is defined as the element-wise square difference between A k and A 0 , i.e., Ξ k i j = (a k i j − a 0 i j ) 2 and that for matrices Σ A 0 and Ξ k we introduce, respectively, the permutations p Σ A 0 and p Ξ k of the linear index i ℓ = i • N + j, which make the entries of the matrices ordered from the smallest to the largest.Fig. S4 confirms that, for both subjects, the entries of the matrix Ξ k are comparable with the entries of the matrix Σ A 0 when plotted against the permutated linear index p Ξ k (i ℓ ) and p Σ A 0 (i ℓ ), respectively.Moreover, the square difference between the entries of the matrices A 0 and A k distributes similarly to the entries of the matrix Σ A 0 , i.e., the higher the uncertainty of a specific weight, the larger the change introduced by the optimization algorithm.In this case, linear indices of both matrices are ordered according to the permutation p Σ A 0 .
Figure S3.Dendrograms obtained from the hierarchical clustering applied on matrices X * for subjects 0025428 and 0025447.Red horizontal lines: central levels of the sets L * (level ℓ * 18 for subject 0025428 and level ℓ * 21 for subject 0025447).Nodes on the dendrogram are labeled according to the HOA.
Figure S5 shows the results of the robustness analysis for level ℓ * 18 of subject 0025428 and ℓ * 21 of subject 0025447.As expected, the average comparison measure B between the target partition (derived from experimental data) and the partition obtained by simulating the network with the optimized and perturbed structural connectivity matrices A k becomes lower as σ A grows.It can be observed that B is above the maximum value obtained by simulating the network with the original structural connectivity matrix A 0 in the large region enclosed within the yellow dashed curve.In this region, the optimized network (with connectivity matrix A k ) behaves in better accordance with the observed functional connectivity than the original network (with connectivity matrix A 0 ).This confirms that the model is robust to perturbations in the connection weights.
These results are aligned with the case study presented in the main paper, supporting the reliability of the proposed method.

Figure S2 .
Figure S2.Left panel: Ψ 1 (ℓ) index for all subjects of the HNU1 dataset 1 , color-coded according to the bottom colorbar.Right panels: Ψ 1 (ℓ) curves for subjects 0025428 and 0025447, which show the second and third highest average similarity index Ψ 1 , respectively.Red dots: Ψ 1 (ℓ) peaks, which identify the sets of selected levels L * .

Figure S4 . 3
FigureS4.Entries of the matrices Σ A 0 (red dots) and Ξ k (black dots) for subjects 0025428 (top panels) and 0025447 (bottom panels).Left panels: entries of both Σ A 0 and Ξ k are displayed from the smallest to the largest and are plotted on a semi-logarithmic scale, with their linear indices i ℓ ordered according to permutations p Σ A 0 and p Ξ k , respectively.Right panels: entries of Σ A 0 are displayed on a linear scale from the smallest to the largest, with their linear indices i ℓ ordered according to permutation p Σ A 0 ; entries of Ξ k are displayed following the same permutation.